The production of a \(\mu ^+\mu ^-\)  pair from the scattering of a muon-neutrino off the Coulomb field of a nucleus, known as neutrino trident production, constitutes a sensitive probe of new neutral current interactions in the lepton sector [1]}, [2]}. Generalizing the formula of Ref. [2]} we find
\(\begin{split}&\dfrac{{{\sigma _{\rm SM + NP}}}}{{{\sigma _{\rm SM}}}} = 1\\& + 8 \dfrac{g_{22}^{\ell L}}{g_2^2}\dfrac{m_W^2}{m_{Z^{\prime }}^2} \dfrac{\left( {1 + 4s_W^2} \right)\left( {g_{22}^{\ell L} + g_{22}^{\ell R}} \right)+ \left( {g_{22}^{\ell L} - g_{22}^{\ell R}} \right)}{\left( {1 + 4s_W^2} \right)^2 + 1}\,.\end{split}\) 
